With each succeeding generation of integrated circuit technology, variability is proportionately increasing. The sources of such variability include manufacturing variations, device fatigue, environmental variations and phase-locked loop (PLL) variations. In the case of manufacturing variations, the front-end-of-the-line (FEOL) which are the layers that define the active transistors show variation in the transistor's electrical characteristics. Physical quantities such as the length of the gate, depth of the semiconductor junction or thickness of the oxide cannot be perfectly controlled during manufacturing and hence show variations, which lead to variations in the behavior of the transistors. As the physical dimensions get smaller in modern technologies, variability is proportionately increasing. In addition, the back-end-of-the-line (BEOL), which consists of the metal interconnect layers, also exhibits variability. For example, the thickness, width and inter-layer dielectric thickness of each metal layer are sources of variability. These in turn cause the wires to change their delay, and in fact these sources of variability can change the delay of gates which are driving them and gates which are driven by them.
The second main type of variations is due to device fatigue effects such as hot electron and negative bias temperature instability (NBTI). After a long period of use in the field, transistor characteristics change due to these physical phenomena, leading to changes in the delay of circuit components.
The third main type of variations is due to environmental effects such as temperature and power supply voltage.
The fourth main type of variations is PLL variations which can include PLL jitter and duty-cycle variability.
It is to be noted that in addition to the above, there are other sources of variation such as model-to-hardware miscorrelation, silicon-on-insulator (SOI) history effects and coupling noise. These other types of variation can also be considered during statistical timing analysis of digital integrated circuits.
The variation of delays shown by gates and wires in an integrated circuit can be classified in many different ways. The variation may be from batch-to-batch during the manufacturing, wafer-to-wafer, chip-to-chip or within a single chip. Lens aberration effects during photolithography, for example, can cause variation of the effective length of transistors across a reticle field. There can be temperature and power supply voltage variations across a chip. The variations can also be classified by the time scales during which variability develops. For instance, fatigue effects cause variability over a period of years, whereas across the chip temperature or power supply gradients can develop over seconds or milliseconds, and coupling noise variations can occur in nanoseconds or picoseconds. Whichever way they are classified, it is abundantly clear that these sources of variation are making integrated circuit analysis and design more difficult and must be accurately accounted for during timing analysis.
The traditional timing methodology to handle such variability is to conduct multiple static timing analyses at different “cases” or “corners” to determine the spread of performance of the circuit under these variations. Corners may include, for example, “best case,” “nominal” and “worst case.” Unfortunately, the traditional methodology is breaking down because the number of independent and significant sources of variation is numerous, and too many timing runs would be required. One way to combat this is to worst-case or guard-band against some sources of variation, but this causes pessimism in the performance prediction. Another way to combat the explosion of timing runs required is to skip the analysis at certain corners, but this is risky since the performance of the circuit may be unacceptable at the skipped corners and this may be manifested by chips failing on the tester or in the field. Because of these effects, traditional timing methodologies are rapidly becoming burdensome, as well as risky and pessimistic at the same time.
A solution to the problems faced by traditional timing methodologies is statistical or probabilistic timing analysis. In such an analysis, timing quantities such as delays, arrival times and slacks are not treated as single numbers, but rather as probability distributions. Thus the full probability distribution of the performance of the circuit under the influence of variations is predicted by a single timing run. The problems of unnecessary risk, excessive timing runs and pessimism are all potentially avoided. Four examples of such statistical timing methods in the prior art include Liou et al [J-J. Liou, K-T. Cheng, S. Kundu and A. Krstic, “Fast statistical timing analysis by probabilistic event propagation,” Proc. Design Automation Conference, June 2001, Las Vegas, Nev., pages 661–666], Scheffer [L. Scheffer, “Explicit computation of performance as a function of process variation,” Proc. ACM/IEEE workshop on timing issues in the specification and synthesis of digital systems, December 2002, Monterey, Calif., pages 1–8], Gattiker et al [A. Gattiker, S. Nassif, R. Dinakar and C. Long, “Timing yield estimation from static timing analysis,” Proc. IEEE International Symposium on Quality Electronic Design (ISQED), 2001, pages 437–442] and Jess et al [J. A. G. Jess, K. Kalafala, S. R. Naidu, R. H. J. M. Otten and C. Visweswariah, “Statistical timing for parametric yield prediction of digital integrated circuits,” Proc. Design Automation Conference, June 2003, Anaheim, Calif., pages 932–937]. The references cited above are herein incorporated by reference in their entirety.